Physica A 387 (2008) 6190–6200

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Physica A

journal homepage: www.elsevier.com/locate/physa

Analyzing open-source software systems as complex networksXiaolong Zheng a, Daniel Zeng a,b,∗, Huiqian Li a, Feiyue Wang a,c

a The Key Lab of Complex Systems and Intelligence Science, Institute of Automation, Chinese Academy of Sciences, Chinab Department of Management Information Systems, The University of Arizona, USAc Department of Systems and Industrial Engineering, The University of Arizona, USA

a r t i c l e i n f o

Article history:Received 15 January 2008Received in revised form 9 June 2008Available online 6 July 2008

Keywords:Open-source software systemsComplex networksDegree distribution

a b s t r a c t

Software systems represent one of the most complex man-made artifacts. Understandingthe structure of software systems can provide useful insights into software engineeringefforts and can potentially help the development of complex system models applicableto other domains. In this paper, we analyze one of the most popular open-source Linuxmeta packages/distributions called the Gentoo Linux. In our analysis, we model softwarepackages as nodes and dependencies among them as edges. Our empirical results showthat the resulting Gentoo network cannot be easily explained by existing complex networkmodels. This in turnmotivates our research in developing twonewnetwork growthmodelsinwhich a newnode is connected to an old nodewith the probability that depends not onlyon the degree but also on the ‘‘age’’ of the old node. Through computational and empiricalstudies, we demonstrate that our models have better explanatory power than the existingones. In an effort to further explore the properties of these new models, we also presentsome related analytical results.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

In the past few years, we have witnessed dramatic growth in the body of the literature studying complex networksin a wide variety of fields. Examples include the movie actor collaboration [1,2], science collaboration [3], WWW [4–6],Internet [7,8], protein–protein interaction networks [9], transportation networks [10], and e-mail networks [11]. Softwaresystems [13–18] represent another important class of systems that can be studied using the complex network analysisframework. Such systems are one of themost complexman-made artifacts and play a critically important role in themodernsociety. The U.S. National Institute of Standards and Technology (NIST) estimated in 2002 that software faults alone can costthe U.S. economy $59.5 billion annually, representing about 0.6% of the U.S. gross domestic product [12]. Understanding thestructure of software systems can potentially lead to better software engineering practice and increased system reliability.In particular, complex network theory can contribute to the development of quantifiable measures of software structurethat in turn can help to develop large software systems with desirable structural designs and identify potential structuralproblems.

Research on studying software from the perspective of complex systems is emerging but to date has not receivedwide-spread attention [13]. We suspect that the main reasons behind this are the difficulties with data collection andthe lack of applicable models [14]. In the past few years, however, the fast growth of open-source software has allowedresearchers to collect data easily. In addition, the recent complex network literature is also making an increasingly rich setof models available to analyze software systems. In this paper, we focus on an analysis of software systems using complex

∗ Corresponding address: Department of Management Information Systems, The University of Arizona, McClelland Hall 430K, Tucson, USA.E-mail address: zeng@email.arizona.edu (D. Zeng).

0378-4371/$ – see front matter© 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2008.06.050

X. Zheng et al. / Physica A 387 (2008) 6190–6200 6191

Fig. 1. The Gentoo network.

network theory. A software system is composed ofmany interacting units and subsystems and the interactions among themdirectly reflect the design, coding, and execution of software. As such, analyzing andmodeling software systems as complexnetworks can afford us meaningful understanding regarding the formation and evolution of code-based software structuresand the processes governing the development of software systems. Furthermore, understanding the functional organizationof evolving software systems may also provide new insights into network growth in a significant engineering setting thatcould potentially help us to gain better understanding of other classes of complex networks.

In this paper, we analyze one of the most popular Linux meta packages/distributions called the Gentoo Linux operatingsystem. In Gentoo Linux, software applications are distributed in the form of packages. As is common in open-sourcesoftware development, the developers of many software applications rely on using other open-source packages. Suchdependencies among packages canmanifest themselves in different forms. For instance, one package needs the source codeof another package to compile correctly. In another form, source-code dependencies do not exist; rather, the binary-levellibrary sharing is required for a package to function properly. Such software package dependencies often span across manydifferent projects. A package management system serves the purpose of managing such dependencies, which is importantfor both functioning andmaintenance (e.g., automated updating) of software packages. The Portage system is a sophisticatedpackage management system used by Gentoo Linux fashioned after the well-tested package management system used inanother major open-source project—the Berkeley Software Distribution (BSD). Gentoo’s portage manages more than 10000packages and has explicit information about dependencies among packages. Such dependencies represent to a large degreethe true structure of a large software.

In our research, we study the package dependencies existing in Gentoo Linux. By treating packages as nodes and theirdependencies as edges, we construct the Gentoo network.The rest of this paper is organized as follows. In Section 2, wepresent an empirical analysis of the Gentoo network. As some key properties of this network cannot be explained bythe existing network evolution models, we have been motivated to develop our own models. In Section 3, we discussthree related known models and then propose two new evolution models, in which a new node is connected to an oldnode with the probability that depends not only on the degree but also on the age of the old node. Section 4 presents acomputational study based on these models. The results of this study are compared against the findings from the real-world Gentoo network. These simulation results show that our models have better explanatory power than the existingmodels. In Section 5, we present some analytical results related to our new models. Finally, Section 6 concludes the paperby summarizing our findings and discussing possible future research.

2. An empirical analysis of an open-source software dependence graph

Linux is an open-source operating system that has a rich set of application software packages. Gentoo Linux is one of themost popular Linuxmeta package distributionswith a flexible and powerful softwaremanagement facility called the Portagesystem. The Portage system explicitly tracks dependencies among software packages, making software installation andupgrading automatic and efficient. The Gentoo Linux has been quite popular in both its user base and developer communitysince its inception, in part due to the power and flexibility of Portage. The number of the software packages in GentooLinux has been growing very quickly in the past few years. Until February 2007, Gentoo’s Portage has managed over 16800packages. Most of the data used in this paper were collected from the Portage and the Gentoo Web site [19]. Note that ourdata are quite different from those used in previous studies concerning software. In [13], the authors analyze small-scalesoftware systems (e.g., DM consisting of 187 packages, VTK 788 packages). Data used in [14] are mainly concerned withsoftware change events from revision histories or logs and do not capture package dependencies.

Fig. 1 illustrates what the Gentoo package dependency network (referred to as the Gentoo network thereafter) lookslike, where the nodes represent the packages available from Gentoo and the edges the dependencies among packages. Forexample, nodes labeled as 3dfb-0.6.1 and glib-2.8.5 represent packages 3dfb-0.6.1 and glib-2.8.5 respectively; whereas theedge between nodes 3dfb-0.6.1 and glib-2.8.5 indicates the dependency between these two packages.

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Table 1summary of various symbols for graph measures

Symbol Measure

Nn Number of nodes in the networkNe Number of edges in the networkk DegreeR Degree growth rateM TheMth monthP(k) Degree distribution of a node with degree kC Clustering coefficient

Fig. 2. Plot of the degree distribution of the Gentoo network.

In this section, we make brief observations concerning the properties of the Gentoo network. Some symbols used inthis section are summarized in Table 1. We mainly focus on degree distribution but also discuss the sparsity, clusteringcoefficient, degree growth rate, and node growth of the network.

Sparsity

As of February 2007, the number of nodes Nn in the Gentoo network is 16803, and the average degree 〈k〉 is about2.7576. The total number of edges Ne is 23 168 while the total number of edges of the complete graph with 16803 nodes isC216 803 = (16 803 × 16 802)/2 = 141 162 003. The ratio between them is about 0.016%. The Gentoo network is sparse in

general.

Clustering Coefficient

The clustering coefficient Ci of the node i is defined as 2ei/ki(ki−1), where ei is the number of the existing edges betweenthe ki neighbors of node i. For nodes with degree 0 or 1, we set Ci = 0. The clustering coefficient C for the entire networkis the average of Ci over all the nodes [20]. The clustering coefficient C of the Gentoo network is about 0.0318 while theclustering coefficient of a corresponding random graph Crand = 〈k〉/Nn = 0.0001641. That is, the clustering coefficient ofthe Gentoo network is about 194 times higher than that of the random graph.

Degree distribution

Degree distribution in random graphs is either binomial or Poisson when the size of the graph is large. However, manyreal-world networks have been found to follow different patterns. For instance, many networks’ degree distribution followsthe power-law property, while others’ exhibits non-power-law features such as exponential cutoffs [21–24]. Fig. 2 showsthe degree distribution of the Gentoo network in log k to log P(k) view. Note that log in this paper is natural logarithm. Theplot shows that the degree distribution of the Gentoo network does not follow the Poisson distribution and a pure power-law distribution. Rather they closely resemble a stretched exponential distribution because the curve declines more rapidlythat the dashed line with a slope 2.3391. However, the tail of the curve appears to be much longer than that of a stretched

X. Zheng et al. / Physica A 387 (2008) 6190–6200 6193

Fig. 3. Average growth rate as a function of the time.

exponential distribution. Amaral et al. [21] and Albert et al. [25] have found this similar phenomenon in other real-worldnetworks and suggested that the aging effect of the nodes can be a leading explanation.

Degree growth rate

To further illustrate the impact of the age of nodes on their attractiveness (in terms of being used by a new package),we conduct an empirical analysis based on degree growth rate for a selected set of 539 packages/nodes. These packages arethose that are already recorded in Gentoo’s portage system by January 2003 and have been kept as an active part of Gentoountil at least February 2007. We observe how these nodes’ degrees change on average over this 50-month time window.To quantify the degree growth rate for month t for package i, we use Ri(t), defined as (ki(t) − ki(t − 1))/ki(t) where ki(t)denotes the degree of i at month t . Fig. 3 plots the average degree growth rate as a function of the time. This plot indicatesthat the average degree growth rate of these packages exhibits an interesting two-phase behavior. In the first phase, therate grows and reaches a maximum point whereas in the second phase, the rate begins to decrease slowly. This two-phasebehavior strongly suggests the presence of the aging effect and has an intuitive explanation. As Gentoo Linux grows, someimportant software packages are heavily utilized in new packages, exhibiting the ‘‘winner takes all’’ phenomenon. However,with the rapid increase of available software packages in Gentoo Linux as shown in Fig. 4, the number of packages that canprovide similar functions or capacities as those popular packages also increases. As a result, more choices become available,potentially reducing the attractiveness of those high-degree packages.

3. Model development

As shown in our empirical analysis presented in the previous section, the Gentoo network is sparse and has a highclustering coefficient. In addition, some nodes have very large degrees and the degree distribution has a large tail. Thissection presents our work attempted to develop models to explain the evolution of the Gentoo network. We start with abrief review of some related known models and then present our models.

3.1. Related models

In the past few years, a large number of network models [26–34] have been developed. Some of these models areextensions to the classic BAmodel [2], which is based on twomechanisms: incremental growth and preferential attachment.We discuss briefly three models relevant to our study, namely the BA model, the Krapivsky–Redner–Leyvraz (KRL)model [27], and the Dorogovtsev–Mendes (DM) model [28].

BA modelIn this model, the network starts with an initial set of nodes. At each time the network grows with the addition of new

nodes. For each newly added node, new edges are added between it and some old nodes. The nodes to receive new edgesare chosen following a linear preferential attachment rule, that is, the probability 5(k) of an old node receiving a new edgeis proportional to its degree k,

5(k) ∼ k. (1)

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Fig. 4. The Node growth.

When k is sufficiently large, the degree distribution P(k) follows a power-law dependence P(k) ∝ k−γ with a fixedexponent. The BA model captures a mechanism that can result in the power-law degree distribution. On the other hand, itpredicts a power-law degree distribution with a fixed exponent, whereas the exponents measured for many real networkscan vary between 2.1 and 4. In addition, the degree distribution of real networks can have non-power-law features such asexponential cutoffs or saturation for small k [21–25].

KRL modelThe model is proposed by Krapivsky, Redner, and Leyvraz [27]. Replacing the BA model’s linear preferential attachment,

they use a nonlinear preferential attachment rule. When choosing the nodes to which a new node connects, the probability5(k) depends on kα ,

5(k) ∼ kα, (2)

where α is a tunable parameter.

DM modelThe BA model and the KRL model have one common characteristic: their preferential attachment rules depend only on

the degree of the old node. However, in applications such as reference networks aging occurs: the authors rarely cite veryold papers. Dorogovtsev and Mendes proposed an extended model in which the probability 5(k) is dependent not only onthe degree k of the old node but also on its age τ , that is,

5(k) ∼ kτ−β , (3)

where β is a tunable parameter.

3.2. Two new models

Based on our empirical observations, we note that in the Gentoo network some nodes have a clear finite lifetime.Standard models such as the BA and the KRL do not model such aging effect. See Fig. 5(a) and (b) for simulation resultsshowing the unsuitability of these models in explaining the Gentoo network. The DM model explicitly considers not onlythe node’s degree but also its age. However, there is still a large gap between what the model predicts and the real-worldobservations as shown in Fig. 5(c). Part of the reason seems to be that when the nodes are very ‘‘young,’’ the DM modelshows approximately linear preferential attachment whereas many real-world networks including the Gentoo networkdisplay nonlinear preferential attachment.

We propose two new models to explain the evolution of networks exemplified by the Gentoo network: the DegreeDependent adjustable Evolution with Aging (DDEA) model and the Degree and Age dependent Adjustable Evolution (DAAE)model. These twomodels extend theKRL andDMmodels by following similar growthpatterns butwith different preferentialattachment rules. In DDEA and DAAE, the probability that a new node is connected to an old node is not only proportionalto the degree k of the old node but also dependent on the age τ of the old node. Details of these two models are as follows.

X. Zheng et al. / Physica A 387 (2008) 6190–6200 6195

Fig. 5. Comparison of the degree distributions in log–log plots. (a) Simulation results of the BA model. (b) Simulation results of the KRL model, forα = 1.22, 1.32, 1.45 respectively. (c) Simulation results of the DMmodel, for β = 0.5, 0.05, 0.005 respectively.

Network growthWe observe that the Gentoo network starts with a small number of nodes and then the number of nodes in the network

grows very rapidly. Both DDEA and DAAE start with a small number (m0) of connected nodes. Subsequently, at each timestep we add a new node with m edges that link this new node to m existing nodes. Note that the number of newly addededgesm is not fixed. Rather it is drawn from a known discrete distribution.

Preferential attachmentIn the Gentoo network, some nodes have very high degrees, corresponding to some popular software packages based on

which many other applications have been developed. This suggests preferential attachment based on the degree of the oldnode. Also we often observe that with the growth of Gentoo Linux, new software packages become available with a set offunctionalities extending (or reimplementing) those offered by the existing popular packages. Soon after that, developersare starting to use these new packages and the popularity of the old ones starts to drop. This substitution effect leads to ourmodeling effort that adjusts the preferential attachment probabilities based on both the degree and the age of an node. Thepreferential attachment rules of DDEA and DAAE follow different forms.

(1) In DDEA, the attachment probability 5(k) is given by

5(k) ∼ kαe−βτ , (4)

where τ is equal to t − sk, with t denoting the current time and sk the time the node kwas born. (We dropped the subscriptdenoting the node index from τ to simplify the notation.) Parametersα andβ are tunable. In thismodel, a young high-degreenode has a high chance to be connected to a new node. The aging effect, however, will make the node less attractive as timeprogresses even if it is already well-connected.

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Fig. 6. Comparison of the degree distributions in log–log plots. (a) Simulation results of the DDEA model, for α = 1.24 and β = 0.0006. (b) Simulationresults of the DAAE model, for α = 2.14 and β = 0.00073.

(2) In DAAE, 5(k) takes the following form:

5(k) ∼ kατ−βk. (5)

Similar to DDEA, DAAE ensures that a young, well-connected node will have high probability to be connected to a new nodeand that aging will reduce the connection probability. The key difference between these twomodels is that in DAAE a youngnode not yet well-connected will have a higher probability to be connected to a new node.

From amodeling perspective, we hypothesize that DDEA and DAAE can be potentially fruitfully applied to describe othersimilar networks whose nodes have a finite lifetime (e.g., reference networks) or whose edges have finite capacity (e.g., theInternet router network). Such studies are beyond the scope of this paper.

4. A computational study

Based on the models described in Section 3, we have conducted a computational study. The simulation results arecompared with the actual observations made from the Gentoo network. The simulation procedure governing the initialnetwork setup and subsequent growth closely resembles the experimental method used by Liu et al. [6]: We start with twoconnected nodes. At each step of network growth, a new node withm edges will be connected tom different old nodes untilthe total number of nodes reaches the number of actual nodes in the Gentoo network. Herem is not a constant but a randomvalue drawn from the set {1, 2}with corresponding probabilities {p1, p2}. These twoprobabilities are estimated by equationsN1 +N2 = Nnew and N1 +2N2 = Enew, where Nnew denotes the total number of newly added nodes, Enew the total number ofnewly added edges, and N1 and N2 the numbers of the newly added nodes with one edge and two edges, respectively. Sincethe total numbers of nodes and edges in the Gentoo network are 16803 and 23168 respectively, in our simulations, Nnew

is set to 16801 and Enew to 23167. We calculated p1 =N1

Nnew= 0.6211 and p2 =

N2Nnew

= 0.3789. Because of the stochasticnature of the network growth, each simulation setup was repeated for 30 times and the results are the averages over these30 replications. Fig. 5 summarizes the simulation results of the BA, KRL, and DM models. Fig. 6 summarizes the simulationresults based on our DDEA and DAAE models.

Fig. 5(a) shows the degree distributions of the BAmodel and the Gentoo network.We note that the degree distribution ofthe Gentoo network deviates significantly from that produced by the BAmodel. The absolute value of the slope of the Gentoonetwork is larger than that of the BA model. The Gentoo network’s degree distribution is less smooth than the BA model’sand has a significantly longer tail. The clustering coefficient produced by BA model is 0.000261, which is far smaller thanthat of the observed value 0.0318. Fig. 5(b) compares the degree distributions of the KRL model and the Gentoo network.In the case of α = 1.32, the degree distribution of the KRL model provides a good match for the Gentoo network when thedegree is not large. However, as the degree increases, the differences start to show. The maximum degree of the KRL modelsimulation results is far larger than that of the Gentoo network. The clustering coefficient in this case is 0.2361, which ismuch larger that the actual value. When α is equal to other values, we also cannot obtain good results. For example, when αis equal to 1.45, in this case, the clustering coefficient 0.3440 and the degree distribution are different from the actual value.When α is equal to 1.22, though the clustering coefficient is 0.0584, which is close to the actual one. However, the degreedistribution in this case significantly deviates from the actual curve. Fig. 5(c) shows the degree distributions of the DMmodeland the Gentoo network. We can clearly see their differences. When the parameter β is equal to 0.5, 0.05, and 0.005, theabsolute values of the slope of the DM model are smaller than that of the actual curve and the tails are much shorter. The

X. Zheng et al. / Physica A 387 (2008) 6190–6200 6197

resulting clustering coefficients are equal to 0.0002447, 0.0008696, and 0.0016. All of these three values are much smallerthan that of the actual value. Setting β to other values seems to reduce the overall descriptive power of the DMmodel. Thegeneral observation is that none of these three models provide a good explanation of the observed properties of the Gentoonetwork.

The degree distributions of the DDEA and DAAE models are shown in Fig. 6 (a) and (b), respectively. From Fig. 6(a) weobserve that the simulation results of the DDEA model are similar to the actual observations in general, indicating that theDDEA gives a better description of the data than the three existing models examined before. We also note the remainingproblems: The maximum node degree of the networks generated by DDEA is significantly larger than that of the actualGentoo network. The clustering coefficient in this case is 0.1801, which is a little larger than that of the actual value. Weexperimented with various settings of α and β . It is possible to obtain smaller maximum degree but always at the expenseof much poorer descriptive power with respect to other aspects of the degree distribution. Also the clustering coefficientsare different from the actual value. From Fig. 6(b), we observe that the simulation results of the DAAE model mimic theactual network very well. The slope of the DAAE’s degree distribution is very close to that of the Gentoo network when thedegree is not so large. The tail end of the DAAE degree distribution also closely resembles that of the Gentoo network withsimilarmaximumdegrees. The clustering coefficient produced by our DAAE is 0.0381, which is very close to the actual value.Overall, the DAAE model provides better characterization of the Gentoo network than other models in multiple dimensionsand could provide useful insights into the evolution of Gentoo Linux and networks alike.

5. Analytical properties of the DAAE model

In this section, we derive some useful analytical properties of the DAAE model. These results help to frame some of thecomputational findings discussed in Section 4 in a formal setting. The technical approach adopted here is largely based onthe effective medium approach used by Dorogovtsev and Mendes [28]. Since the results for the exponents do not dependon the number of edges added to the network each time [35], we consider exclusively network growth with one new nodeand one edge.

We observe that P(k, s, t), the probability that the degree of node s at time t is equal to k, can be obtained by the δ-function, i.e. P(k, s, t) = δ(k − k̄(s, t)), where k̄(s, t) is the mean degree of node s at time t . We have

∂ k̄(s, t)∂t

=k̄(s, t)α(t − s)−β k̄(s,t)∫ t

0 duk̄(u, t)α(t − u)−β k̄(u,t), (6)

where k̄(0, 0) = 0, k̄(t, t) = 1. We apply∫ t0 ds to Eq. (6) and obtain∫ t

0k̄(s, t)ds = 2t. (7)

In order to solve Eq. (6), we let

k̄(s, t) = κ(ξ), (8)

where ξ = s/t . From Eqs. (7) and (8), we obtain∫ 1

0κ(ξ)dξ = 2. (9)

Based on Eqs. (6) and (8), we derive

−dκ(ξ)

dξξ =

κ(ξ)α(t − tξ)−βκ(ξ)∫ 10 κ(ζ )α(t − tζ )−βκ(ζ )dζ

(10)

and

κ(1) = 1. (11)

Using the fact∫ 1

0κ(ζ )α(t − tζ )−βκ(ζ )dζ = γ , (12)

we rewrite Eq. (10) as

−dκ(ξ)

dξξ =

1γ

κ(ξ)α(t − tξ)−βκ(ξ). (13)

We now give two properties.

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Property 1. If α 6= 1, and β → 0, then at time t, degree distribution P(k, t) decreases as a stretched exponential in k, such that:

P(k, t) ∼ k−αeγ

1−α(1−k1−α). (14)

Proof. If α 6= 1, Eq. (13) can be transformed into

dκ(ξ)1−α= −

1 − α

γ ξ(t − tξ)−βκ(ξ). (15)

As it is difficult to integrate Eq. (15) directly, we apply an approximation scheme using Taylor’s formula to expand(t − tξ)−βκ(ξ).

dκ(ξ)1−α u −1 − α

A1γ ξ [1 + M1(ξ − ξ0)]dξ, (16)

where

A = (t − tξ0)βκ(ξ0), (17)

M = β

[κ ′(ξ0) ln(t − tξ0) −

κ(ξ0)

1 − ξ0

]. (18)

We now solve Eq. (16) and obtain

κ(ξ)1−α u −1 − α

Aγ (1 − Mξ0)ln

[ξ

1 + M(ξ − ξ0)

]+ C1. (19)

Recalling Eq. (11), we can determine constant C1. The final solution is

κ(ξ)1−α u −1 − α

Aγ (1 − Mξ0)ln

[ξ [1 + M(1 − ξ0)]

1 + M(ξ − ξ0)

]+ 1. (20)

If β → 0, from Eqs. (17), (18) and (20) we can see that

κ(ξ) ∼

(1 −

1 − α

γln ξ

) 11−α

. (21)

Since P(k, t) ∝ ∂s/∂k, we conclude

P(k, t) ∼ k−αeγ

1−α(1−k1−α).

Property 2. If α = 1, and β → 0, then at time t, degree distribution P(k, t) shows the power-law property and follows

P(k, t) ∼ k−3. (22)

Proof. If α = 1, Eq. (13) can be transformed into

d ln κ(ξ) = −1

γ ξ(t − tξ)−βκ(ξ)dξ . (23)

To obtain an approximate solution, we apply Taylor’s formula to (t − tξ)−βκ(ξ) and obtain,

d ln κ(ξ) u −1

Aγ ξ [1 + M(ξ − ξ0)]dξ . (24)

The solution to Eq. (24) is

ln κ(ξ) u −1

Aγ [1 − Mξ0]ln

[ξ

1 + M(ξ − ξ0)

]+ C2. (25)

Recalling Eq. (11), we can determine constant C2 and get

ln κ(ξ) u −1

Aγ (1 − Mξ0)ln

[ξ [1 + M(1 − ξ0)]

1 + M(ξ − ξ0)

]. (26)

X. Zheng et al. / Physica A 387 (2008) 6190–6200 6199

If β → 0, we obtain

ln κ(ξ) ∼ −1γ

ln ξ . (27)

Recalling Eqs. (9) and (12), we obtain

γ → 2. (28)

Then

κ(ξ) ∼ ξ−12 . (29)

Finally, we obtain the degree distribution

P(k, t) ∝ ∂s/∂k ∝ k−1−γ∼ k−3. � (30)

From the analysis above, in the DAAE model, we have found that if the parameter α is not equal to 1 and β is verysmall, then the degree distribution decreases as a stretched exponential. However, if the parameter α is equal to 1 and β issufficiently small, then the degree distribution of this model exhibits the power-law property.

6. Concluding remarks

In this paper, we present an empirical analysis of the Gentoo network. We have analyzed network properties includingdegree distribution, sparsity, clustering coefficient, degree growth rate, and node growth. As existingmodels do not providesatisfactory explanation for the observed characteristics concerning theGentoo network’s degree distribution and clusteringcoefficient, we have developed and evaluated two new network growth models, DDEA and DAAE, based on the KRL and DMmodels. By comparing simulation results with empirical observations, we demonstrate that these newmodels, in particular,the DAAE model, are able to reproduce the observed properties of the Gentoo network better than their predecessors. TheDAAEmodel has two notable features. (a) One component of the preferential attachment probability is based on a nonlinearterm of the node degree. (b) The exponent of the aging-related term is dependent on the node degree.

For future research, additional evaluation of theDDEA andDAAEmodels needs to be conducted, in both software researchand other application domains. As to open-source research,work is being pursued by several groups to analyze the developernetwork [15–18]. It will be interesting to study the co-evolution between developer and software package networks. Fromboth engineering (package dependency) and management (developer network) perspectives, investigating intrinsic fitnessmeasures for each node as in Ref. [33] and incorporating such measures in model development for open-source softwarepresent many interesting research opportunities. As pointed out in Ref. [13], investigating software systems from a complexnetwork perspective helps one to gain better understanding of relationships among software network structure, objectcomplexities, object interactions, development processes, and system evolution. Study a number of open-source softwaresystems beyond Gentoo Linux and developing models and measures that could be applicable to open-source softwareengineering in general could lead to fruitful research contributions.

Acknowledgments

We thank the reviewers for the valuable comments.We would like to thank Liang Sun, Wei Cui, Feng Zhang, and HongMo, for useful discussions and helpful suggestions. This work was supported by the National Natural Science Foundation ofChina under Grants. 60621001 and 60573078, by the Chinese Academy of Sciences through an international collaborationGrants 2F05NO1 and 2F07C01, and by theMinistry of Science and Technology under the 973 programGrants. 2006CB705500and 2004CB318103. The second author wishes to acknowledge support from the U.S. National Science Foundation throughGrants IIS-0527563 and IIS-0428241.

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